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07 Nov 2023, 08:42
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If \(m\) and \(n\) are positive integers, and the remainder when \(m\) is divided by \(n\) is equal to the remainder when \(n\) is divided by \(m\), then which of the following could be the value of \(m*n\)?
I. 12
II. 24
III. 36
A. I only
B. II only
C. III only
D. I and III only
E. II and II only
I. 12
II. 24
III. 36
A. I only
B. II only
C. III only
D. I and III only
E. II and II only
07 Nov 2023, 08:42
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Official Solution:
If \(m\) and \(n\) are positive integers, and the remainder when \(m\) is divided by \(n\) is equal to the remainder when \(n\) is divided by \(m\), then which of the following could be the value of \(m*n\)?
I. 12
II. 24
III. 36
A. I only
B. II only
C. III only
D. I and III only
E. II and II only
We are given that the remainder when \(m\) is divided by \(n\) is equal to the remainder when \(n\) is divided by \(m\). This implies that \(m\) must be equal to \(n\), and consequently, \(mn\) must be the square of an integer. Therefore, the answer is C, III only.
To further elaborate on why \(m\) must be equal to \(n\), assume \(m < n\). In this case, the remainder when \(m\) is divided by \(n\) would be \(m\) itself. However, since we are given that the remainder when \(m\) is divided by \(n\) is equal to the remainder when \(n\) is divided by \(m\), it would mean the remainder when \(n\) is divided by \(m\) should also be \(m\). This scenario is impossible because the divisor must always be greater than the remainder. A similar contradiction arises if we assume \(n < m\). Therefore, the only consistent possibility is that \(m\) equals \(n\).
Answer: C
If \(m\) and \(n\) are positive integers, and the remainder when \(m\) is divided by \(n\) is equal to the remainder when \(n\) is divided by \(m\), then which of the following could be the value of \(m*n\)?
I. 12
II. 24
III. 36
A. I only
B. II only
C. III only
D. I and III only
E. II and II only
We are given that the remainder when \(m\) is divided by \(n\) is equal to the remainder when \(n\) is divided by \(m\). This implies that \(m\) must be equal to \(n\), and consequently, \(mn\) must be the square of an integer. Therefore, the answer is C, III only.
To further elaborate on why \(m\) must be equal to \(n\), assume \(m < n\). In this case, the remainder when \(m\) is divided by \(n\) would be \(m\) itself. However, since we are given that the remainder when \(m\) is divided by \(n\) is equal to the remainder when \(n\) is divided by \(m\), it would mean the remainder when \(n\) is divided by \(m\) should also be \(m\). This scenario is impossible because the divisor must always be greater than the remainder. A similar contradiction arises if we assume \(n < m\). Therefore, the only consistent possibility is that \(m\) equals \(n\).
Answer: C
24 Jan 2025, 03:20
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This is an important concept! Thanks, really a high-quality question
